
where \(\bar{\partial } = \partial _{\bar{z}} = \frac{1}{2}(\partial _1 + i \partial _2)\) . Note that the differential operators in \(\mathcal {D}(x)\)  contains only \(\bar{\partial }\) , i.e. \(\partial _z\)  is absent. Since we already showed that \(\Phi _{K_m, B}\)  is the solution of Eq. REF , \(f(z) \Phi _{K_m, B}\)  is also a solution as long as \(f(z)\)  is a holomorphic function because then \(\bar{\partial } f(z) = 0\) . To construct the solution with the Bloch boundary conditions at an arbitrary momentum \(k\) , introduce[1]}
\(\eta (z = x_1 + i x_2) = \frac{\vartheta _{\frac{k \cdot L_1}{2\pi } - \frac{1}{6}, \frac{1}{6} - \frac{k \cdot (L_2 - L_1)}{2\pi }}\left( \frac{z}{|L_1|} e^{-i \frac{\pi }{6}}, e^{i \frac{2\pi }{3}} \right)}{\vartheta _{-\frac{1}{6}, \frac{1}{6}}\left( \frac{z}{|L_1|} e^{-i \frac{\pi }{6}}, e^{i \frac{2\pi }{3}} \right)} \) 
